Multidimensional extension of the Morse--Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of \ZZ^d definable by a first order formula in the Presburger arithmetic . With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $\ZZ^d$ definable in in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often

Decidability of the HD0L ultimate periodicity problem

In this paper we prove the decidability of the HD0L ultimate periodicity problem

Recurrence along directions in multidimensional words

In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A d-dimensional word is called uniformly recurrent if for all s_1,...,s_d, there exists n such that each block of size (n,…,n) contains the prefix of size (s1,…,sd). We are interested in a modification of this property. Namely, we ask that for each rational direction (q_1,…,q_d), each rectangular prefix occurs along this direction in positions ℓ(q1,…,qd) with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional words satisfying this condition, and more generally, a series of four increasingly stronger conditions. In particular, we study the uniform recurrence along directions of multidimentional rotation words and of fixed points of square morphisms

Multidimensional extension of the Morse--Hedlund theorem

International audienceA celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of \ZZ^d definable by a first order formula in the Presburger arithmetic \langle\ZZ;<,+\rangle. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of \ZZ^d definable in \langle\ZZ;<,+\rangle in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often